#9240964

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,12)(4,14)(5,9)(6,13)(7,8)(10,15)(11,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(2,8)(3,5)(10,14)(11,13)(19,27)(21,31)(23,32)(25,28)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(1,3,7,5)(2,9,8,12)(4,11,15,13)(6,10,16,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,7,15)(2,10,8,14)(3,13,5,11)(6,12,16,9)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,7,3)(2,9,8,12)(4,11,15,13)(6,14,16,10)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32);(1,6)(2,11)(3,10)(4,12)(5,14)(7,16)(8,13)(9,15)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9148160);